Instructors of introductory physics courses expect that students will be able to apply what they have learned in previous mathematics courses to what they are learning in introductory mechanics. Once they have completed introductory mechanics they will be able to apply the conceptual and procedural knowledge developed in mechanics to the learning of electricity and magnetism. Fulfillment of these expectations requires that students transfer knowledge obtained in previous courses to different contexts in subsequent courses. However, there is little agreement among education researchers about the mechanism of transfer from one context to another and about how to measure transfer. This study describes an initial investigation into transfer across contexts in introductory physics, and into the measurement of transfer. The first part of this book describes an investigation we have conducted to better understand the extent to which students transfer knowledge from math to physics, concentrating on the mathematics of proportional reasoning that is ubiquitous in introductory physics courses. Results suggest that many students do not have the level of preparation expected by their instructo